I photographed this big roach one nite last week in my in-laws' backyard in Florida. It was ~3 cm long. A quick search on BugGuide.net identified it as an Australian cockroach (Periplaneta australasiae). The distinguishing characteristic of this species is the yellow border around its thorax.
They were introduced to southern U.S. presumably from Australia, and now appear to have become naturalized at least in Florida. In other words, along with the snail Bradybaena similaris and the myriads of other alien creatures, they are here to stay.
This particular individual was quite docile. Unlike some other cockroaches I am familiar with (for example, the German cockroach, Blattella germanica), it allowed to get get quite close to it and didn't attempt to run away until after I started poking around with a stick to move the grass blades out of the way for a better picture.